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Strong Convergence of Two Proximal Point Algorithms with Possible Unbounded Error Sequences

Author

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  • Behzad Djafari Rouhani

    (University of Texas at El Paso)

  • Sirous Moradi

    (Arak University)

Abstract

We consider a proximal point algorithm with errors for a maximal monotone operator in a real Hilbert space, previously studied by Boikanyo and Morosanu, where they assumed that the zero set of the operator is nonempty and the error sequence is bounded. In this paper, by using our own approach, we significantly improve the previous results by giving a necessary and sufficient condition for the zero set of the operator to be nonempty, and by showing that in this case, this iterative sequence converges strongly to the metric projection of some point onto the zero set of the operator, without assuming the boundedness of the error sequence. We study also in a similar way the strong convergence of a new proximal point algorithm and present some applications of our results to optimization and variational inequalities.

Suggested Citation

  • Behzad Djafari Rouhani & Sirous Moradi, 2017. "Strong Convergence of Two Proximal Point Algorithms with Possible Unbounded Error Sequences," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 222-235, January.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:1:d:10.1007_s10957-016-1028-5
    DOI: 10.1007/s10957-016-1028-5
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    References listed on IDEAS

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    1. B. Djafari Rouhani & H. Khatibzadeh, 2008. "On the Proximal Point Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 411-417, May.
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    Cited by:

    1. Behzad Djafari Rouhani & Vahid Mohebbi, 2020. "Strong Convergence of an Inexact Proximal Point Algorithm in a Banach Space," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 134-147, July.
    2. Gheorghe Moroşanu & Adrian Petruşel, 2019. "A Proximal Point Algorithm Revisited and Extended," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 1120-1129, September.
    3. Behzad Djafari Rouhani & Sirous Moradi, 2019. "Strong Convergence of Regularized New Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 864-882, June.

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